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Inverse problems are inherently ill-posed, suffering from non-uniqueness and instability. Classical regularization methods provide mathematically well-founded solutions, ensuring stability and convergence, but often at the cost of reduced…
Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function,…
Deep learning-based models have demonstrated remarkable success in solving illposed inverse problems; however, many fail to strictly adhere to the physical constraints imposed by the measurement process. In this work, we introduce a…
Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers…
Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require…
We present a novel framework for the reconstruction of 1D composite signals assumed to be a mixture of two additive components, one sparse and the other smooth, given a finite number of linear measurements. We formulate the reconstruction…
One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization…
Learning-based methods have demonstrated remarkable performance in solving inverse problems, particularly in image reconstruction tasks. Despite their success, these approaches often lack theoretical guarantees, which are crucial in…
We investigate continuous regularization methods for linear inverse problems of static and dynamic type. These methods are based on dynamic programming approaches for linear quadratic optimal control problems. We prove regularization…
In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be…
We discuss the possibility to learn a data-driven explicit model correction for inverse problems and whether such a model correction can be used within a variational framework to obtain regularised reconstructions. This paper discusses the…
Regularization plays a pivotal role in integrating prior information into inverse problems. While many deep learning methods have been proposed to solve inverse problems, determining where to apply regularization remains a crucial…
We consider the variational reconstruction framework for inverse problems and propose to learn a data-adaptive input-convex neural network (ICNN) as the regularization functional. The ICNN-based convex regularizer is trained adversarially…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
Theoretical guarantees for the robust solution of inverse problems have important implications for applications. To achieve both guarantees and high reconstruction quality, we propose learning a pixel-based ridge regularizer with a…
Inverse problems are inherently ill-posed and therefore require regularization techniques to achieve a stable solution. While traditional variational methods have well-established theoretical foundations, recent advances in machine learning…
In the context of linear inverse problems, we propose and study a general iterative regularization method allowing to consider large classes of regularizers and data-fit terms. The algorithm we propose is based on a primal-dual diagonal…
Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…
We develop regularisation theory for dynamic inverse problems, solved using online methods with an infinite time horizon. Using concepts of subregularity to treat nonsmooth regularisers, we prove that time-averaged reconstruction errors…
In this article we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods. We review the problem of projection-regularization from a statistical point of view. A basic…